Only an equation!

Algebra Level pending

If x x in the interval [ a , b ] [a,b] satisfies the equation ∣ x − 3 ∣ + ∣ 4 − x ∣ = 1 \displaystyle |x-3|+|4-x|=1

then find the value of a + b a+b .


The answer is 7.

Anik Mandal by Anik Mandal , 21, India

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1 solution

Rishabh Jain
Jul 6, 2016

∣ x − 3 ∣ + ∣ x − 4 ∣ = 1 |x-3|+|x-4|=1

The equation translates to:

{ 2 x − 7 = 1 if x > 4 1 = 1 if 4 ≥ x ≥ 3 7 − 2 x = 1 if x < 3. \begin{cases} 2x-7=1 & \textrm{if } x > 4 \\ 1=1 & \textrm{if } 4\ge x\ge 3 \\ 7-2x =1 & \textrm{if } x < 3. \\ \end{cases}

There are no solutions in first and third case while the equation is satisfied for all x x in second case. Hence x ∈ [ 3 , 4 ] x\in[3,4] .

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